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Inequalities for the differential subordinates of Martingales, harmonic functions and Ito processes

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Title: Inequalities for the differential subordinates of Martingales, harmonic functions and Ito processes
Author(s): Choi, Changsun
Doctoral Committee Chair(s): Ruan, Zhong-Jin
Department / Program: Mathematics
Discipline: Mathematics
Degree Granting Institution: University of Illinois at Urbana-Champaign
Degree: Ph.D.
Genre: Dissertation
Subject(s): Mathematics
Abstract: In Chapter 1 we sharpen Burkholder's inequality $\mu(\vert v\vert\geq1)\leq2\Vert u\Vert\sb1$ for two harmonic functions u and v by adjoining an extra assumption. That is, we prove the weak-type inequality $\mu(\vert v\vert\geq1)\leq K\Vert u\Vert\sb1$ under the assumptions that $\vert v(\xi)\vert\leq\vert u(\xi)\vert, \vert\nabla v\vert\leq\vert\nabla u\vert$ and the extra assumption that $\nabla u\cdot\nabla v$ = 0. Here $\mu$ is the harmonic measure with respect to $\xi$ and the constant 1 $<K<$ 2, found by Davis, is the best constant in Kolmogorov's weak-type inequality for conjugate functions.In Chapter 2 we get norm inequalities. Let ($\Omega,{\cal F},P$) be a probability space with filtration (${\cal F}\sb{n}).$ Let f be a nonnegative submartingale and g be an adapted sequence. Let d be the difference sequence of f and e of $g{:} f\sb{n}=\sum\limits\sbsp{k=0}{n}\ d\sb{k}$ and $g\sb{n}=\sum\limits\sbsp{k=0}{n}\ e\sb{k}, n\ge 0.$ We prove $\Vert g\Vert\sb{p}\le(r-1)\Vert f\Vert\sb{p}$ under the assumption that $\vert e\sb{n}\vert\le\vert d\sb{n}\vert$ for $n\ge 0$ and $\vert{\rm I\!E}(e\sb{n}\ \mid\ {\cal F}\sb{n-1})\vert\le\alpha\vert{\rm I\!E}(d\sb{n}\ \mid\ {\cal F}\sb{n-1})\vert$ for $n\ge 1.$ Here 0 $\le\alpha\le$ 1, 1 $<p<\infty$ are constants, $\Vert f\Vert\sb{p}={\rm sup}\Vert f\sb{n}\Vert\sb{p}$ and $r=\max\{(\alpha+1)p,p/(p-1)\} .$ We also get similar inequalities $\Vert v\Vert\sb{p}\le(r-1)\Vert u\Vert\sb{p}$ and $\Vert\vert Y\Vert\vert\sb{p}\le(r-1)\Vert\vert X\Vert\vert\sb{p}$ where u, v are smooth functions and X, Y are Ito processes.
Issue Date: 1995
Type: Text
Language: English
Rights Information: Copyright 1995 Choi, Changsun
Date Available in IDEALS: 2011-05-07
Identifier in Online Catalog: AAI9624313
OCLC Identifier: (UMI)AAI9624313

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